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A sharp upper bound for the first Dirichlet eigenvalue of cone-like domains

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Abstract

The aim of this paper is to give a complementary upper bound-type isoperimetric inequality for the fundamental Dirichlet eigenvalue of a bounded domain completely contained in a cone. This inequality is a counterpart to the Ratzkin inequality for Euclidean wedge domains in higher dimensions. We also give a new version of the Crooke–Sperb inequality involving a new geometric quantity for the first eigenfunction of the Dirichlet Laplacian for such a class of domains.

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Acknowledgements

The authors gratefully acknowledge the approval and the support of this research study by the Grant No. 7495-SAR-2017-1-8-F from the Deanship of Scientific Research at Northern Border University, Arar, KSA.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Science (Rafha), Northern Border University, Arar, Saudi Arabia

    Abdelhalim Hasnaoui

  2. Department of Mathematics, FST, University of Tunis El Manar, Tunis, Tunisia

    Abdelhalim Hasnaoui

  3. Department of Mathematics, Faculty of Arts and Science (Turaif), Northern Border University, Arar, Saudi Arabia

    Abir Sboui

  4. Department of Mathematics, ISSATM, University of Carthage, Carthage, Tunisia

    Abir Sboui

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  1. Abdelhalim Hasnaoui
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  2. Abir Sboui
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Correspondence to Abdelhalim Hasnaoui.

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Hasnaoui, A., Sboui, A. A sharp upper bound for the first Dirichlet eigenvalue of cone-like domains. Arch. Math. 115, 691–701 (2020). https://doi.org/10.1007/s00013-020-01499-4

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  • DOI: https://doi.org/10.1007/s00013-020-01499-4

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